loudhornsrule-
Unless you live WAY out in the country, on such a large piece of property you dont have to worry about where your bullets will land, I think you would be better off live-trapping the dogs with a box trap or humane snare trap.
A rifle and scope combination of reasonable quality in an effective long-range caliber as well as enough ammo to build some long-range shooting skills will set you back a heckofalot more than a large box trap, which any farmer or rancher worthy of the name can likely throw together for little or nothing out of bits and pieces already in his junk-pile...
Most critically, with trapping, there is zero chance of a stray shot from your rifle landing in a neighbors living room...or taking out a vehicle on an adjacent road...a misplaced shot can be a life-changing event for everyone involved...
With a box trap or humane snare you can sort the curs from the pets, give the pets back to their owners and seek compensation, give them to your local Animal Control department, or dispatch the unwanted dogs quietly with a .22 short in the weapon you already own...
You can avoid the ruckus a leg-hold trap will cause when dog steps in it....as well as avoid hurting animals that are not part of the problem...
I have a box trap for feral pigs built out of salvaged re-bar panels that does double duty for dogs...after the sour corn has done its work on the pigs, the gut-pile from "Porky" will pull in a few dogs...
My 'across the road' neighbors dogs will occassionally dig under the fence, and get a free meal in the trap but tho they will run deer and harrass livestock these are treasured companions to his kids and so are spared....I just walk them back across the road or holler for the boys to come collect em and then patch the hole in the fence...
There have only been a few 'strange' dogs caught that I could not take to Animal Control, as most of the dogs roaming free in my end of the county are "throw-away " pets dumped out on the backroads by city a**holes and accustomed to people....but for the few dogs so wild I could not put a collar and leash on em, my Ruger 10/22 put the unwanted animal down humanly with a head-shot...the ever-present buzzards clean up after me and all thats left in a couple of days is a green spot in my pasture...
I normally feed the Ruger sub-sonic ammo....these rounds eliminate the supersonic "crack" that carries so far and still have enough energy to cycle the action. I have a steel gong target set up behind my barn and my closest neighbor (1/2 mile) hardly ever hears anything...
For even quieter operation I can hand-load .22 shorts and CB caps...those loads are so quiet that an air rifle makes more noise...but they will not cycle the action on the 10/22.
As so many others have noted in this thread, it is a sad fact that sometimes the ultimate remedy is "Shoot, Shovel and Shut Up"...
I hope you can sort your problem out with a minimum of hassel...
Be Careful Out There
Terry